XJ0200040 IlHCbMa B 3HAZ. 2001. N!>5[108] Particles and Nuclei, Letters. 2001. No.5[108] YAK 539.123 MONOCHROMATIC NEUTRINOS FROM MASSIVE FOURTH GENERATION NEUTRINO ANNIHILATION IN THE SUN AND EARTH KM.Belotskya, M.Yu.Khlopova, K.I.Shibaevh °«Cosmion» Centre, Moscow 6MEPI, Moscow Accumulation inside the Earth and Sun of heavy (with the mass of 50 GeV) primordial neutrinos and antineutrinos of the fourth generation and their successive annihilation is considered. The minimal estimations of annihilational fluxes of monochromatic e, jj,, T neutrinos (neutrinos and antineutrinos) with the energy of 50 GeV are 4.1 • 10~6 cm~2-s~1 from the Earth core and 1.1 • 10~7 cm~2-s~x from the Sun core. That makes the analysis of underground neutrino observatory data the additional source of information on the existence of massive stable 4th generation neutrino. It is shown that due to the kinetic equilibrium between the influx of the neutrinos and their annihilation the existence of new [/(l)-gauge interaction of the 4th generation neutrino does not virtually influence the estimations of annihilational e-, fj,-, r-neutrino fluxes. PaccMaTpHBaerca HaKoruieHHe 3eMneft H CojmueM pejiHKTOBbix TayKenhix (c Maccofi 50 FsB) Heii- TPHHO MeTBepToro noKoueHHa c nocneflyiomeH HX aHHaninauHeH. MHHHManbHbie ouemcH ITOTOKOB MO- HOxpoMaTHHecKHX sjieKTpoHHbix, MtooHHbix H Tay-HeHTpHHO (HeHTpHHO H aHTHHefiTpHHo) c 3Heprnefi 50 F3B OT aHHHnciauHH Taacejibix HefiTpHHO cocraBJiaiOT 4,1 • 10~6 CM~2-C-1 H3 ueHTpa 3CMJIH H 1,1 - 10~7 CM~2-C-1 H3 ueHTpa CojiHua, mo aenaer aHanH3 aaHHbix nofl3eMHbix Heih-pHHHSix 06- CepBaTOpHH flOnOJIHHTejibHbIM HCTOHHHKOM HH(bOpMaU.HH O CyilieCTBOBaHHH HeHTpHHO 4-rO nOKOJieHHa. rioKa3aHo, HTO Bcne^cTBHe KHHerHHecKoro paBHOBecHs MOKfly npaneTOM KOCMHnecKHx HCHTPHHO H HX cywecTBOBaHHe HOBOTO £/(l)-KanH6poBOHHoro B3aHMOfleKcTBHa y HefiTpHHO 4-ro noKO- npaKTHnecKH He BUHaeT Ha oueHKy aHHHrajiauHOHHbix noTOKOB e-, n~, T INTRODUCTION The search for a new (fourth) generation of quarks and leptons is one of the most important longstanding problems of high-energy physics. In addition to experiments on accelerators the effective instruments of such search can serve cosmological and astrophysical manifestations of a new generation, for which the existence of a new neutrino (V4) as the lightest, perhaps, stable weakly interacting neutral particle of the 4th generation has special significance. More than 20 years ago it was shown [1] that neutrino with the mass of a few GeV can be a candidate to the role of a cold dark matter particle in the Universe. The modern experimental constraint on the new neutrino mass follows from the experimental data on the Z-boson width: the mass of v$ must be greater than mz/2 w 45 GeV, where mz is the .Z-boson mass. The analysis of possible virtual effects of the 4th family in the particle data constrains the mass of the 4th neutrino in the allowed range around 50 GeV [2]. The results of DAMA experiment on underground search for WIMPs do not exclude WIMP's mass about 50 GeV, that is consistent with the allowed window for the new neutrino mass derived from the particle 6 Belotsky KM., Khlopov M.Yu., Shibaev K.I. data. If the neutrino with such mass existed then the respective neutrino and antineutrino pairs would have to be in equilibrium in the early Universe. After their freeze-out primordial neutrinos would have to retain in the Universe but their contribution into the modern density is estimated to be rather small (O ~ 10~4) [3]. So the relic 4th generation neutrino cannot play the role of the dominant dark matter component and cosmological test of their existence requires the analysis of more refined astrophysical effects. Such analysis should be carried out assuming the presence of the dominant dark matter of another nature. In a series of papers [4-7] the possible experimental (on LEP) [4, 6] and astrophysical [4, 5, 7] manifestations of new heavy stable neutrino existence were revealed. As any other form of nonrelativistic dark matter, the heavy neutrinos must concentrate in galaxies. Under supposing charge symmetry, together with massive neutrinos the equal number of their antineutrinos concentrates in galaxies. That must lead to neutrino-antineutrino annihilation effects. In the work [7] it was shown that neutrino-antineutrino annihilation in halo of our Galaxy could account for galactic diffuse gamma-background recently discovered by EGRET. Such gamma-background can hardly arise from annihilation of a neutralino, the popular candidate to WIMP, predicted in supersymmetric models, due to the Majorana nature of a neutralino. The fluxes on the Earth of particles from heavy neutrino annihilation in the galactic halo were calculated in [4, 5]. The positron, antiproton, gamma fluxes were shown to be most sensitive for experimental probe of such annihilation. The study of other possible astrophysical effects of the 4th neutrino is of evident interest. In many works [8-10] the accumulation of WIMPs (neutralino, neutrino) in the Earth and Sun with their successive annihilation giving the fluxes of known neutrinos were considered. The annihilational e, \i, r neutrinos can be detected by underground neutrino telescopes. In the present work the analogous processes for concretely the 4th neutrino with the mass of 50 GeV are analyzed. The profound signature for such neutrino is the existence of the annihilation channel to the monochromatic neutrino and antineutrino. Also the possibility for heavy neutrino (together with all the other quarks and Ieptons of the 4th generation) to have new interaction is briefly considered. The additional interaction (or even several) appears in realistic variants of superstring theories in low-energy limit. The influence of new interaction on the effects of heavy neutrino annihilation is estimated. 1. CAPTURE OF GALACTIC MASSIVE NEUTRINOS BY THE EARTH AND SUN One estimates massive neutrino (further simply neutrino) number density not perturbed by Earth's and Sun's (gravitating body) gravitational fields supposing the number density distribution in the Galaxy in the form where (n) « 10~11(fi/10~'4) cm"3 is the mean number density in the Universe with Hubble n(r _ Q^ constant h = 0.65; n = -~—.—; r is the distance to galactic centre; r is the characteristic 0 (n) distance of distribution. The ratio K of the neutrino concentration in galactic centre to the average one in the Universe, as shown in [11] for weakly interacting matter, is connected Monochromatic Neutrinos from Massive Fourth 1 with analogous ratio between baryonic (B) and dark matter (DM) densities (in the latter the heavy neutrinos contribute with a small weight) n(r = 0) ( ^ K = = 1Q (n) V (Ps+uu) ) ~ Put 7"o = 1.2 kpc and for solar system take r = rs = 8.5 kpc, then for neutrino number density not perturbed by Sun's and Earth's gravitational fields, that we will indicate by subscript «oo», we obtain noo = £s(n) « 2 • 104{n) « 2 • 1CT7 cm'3. The mean velocity of neutrinos in the Galaxy is taken equal to Woo = 300 km/s, which corresponds to the neutrino kinetic energy T^ = 25 keV. Near gravitating body (in potential field) one can show for a neutrino of fixed energy that the concentration and velocity are in the following relationship n/v = const. (1) Neutrino capture rate by gravitating body will be •Ncapt = Y1 navWlnA dV. (2) A J Here n and TIA are, respectively, the concentrations of incident neutrinos and nuclei1 with atomic number A in Earth's or Sun's matter; a is the cross section of neutrino-nucleus interaction; v is the neutrino velocity; toi is the probability for a neutrino to lose in one collision sufficient energy for capturing by gravitational field; integration is over gravitating body volume. The number density and velocity of incident neutrinos are considered to be «perturbed» by gravitational field of the Sun or Earth (according to (1)), but the change of these characteristics of neutrino flux owing to their interaction with the matter is not taken into account. With the use of Eq. (1), the Eq. (2) can be transformed to the form jV = \E -"2o°o /f av2 n dV. (3) capt Wl A A v°° J The distribution density of Earth's matter will be assumed homogeneous, the other more realistic distributions with its much more complicated form lead to small change of the result. For this distribution the square escape (parabolic) velocity dependent on dimensionless radius x = R/Ro, where RQ is the gravitating body radius, inside the Earth has the form =4E^, (4) where t; E = uscE(l) = H-2 km/s is Earth's surface parabolic velocity. Obviously, Earth's esC e gravitational field does not influence neutrino concentration and velocity. It is essential just 'interaction of neutrino with electrons is negligible. 8 Belotsky KM., Khlopov M.Yu., Shibaev K.I. for factor w\. Also Sun's field near the Earth is not significant. In the present work we will not take into account the possibility of existence of «slow» neutrino component revealed in [10]. According to [10] this population of neutrinos moving along stretched orbit around the Sun appears in the result of weak collisions of background neutrinos with nuclei of the Sun. The role of such population in the processes of neutrino accumulation by the Earth can be important due to large value of wi, that is accounted for much lower velocity of these neutrinos (of the order of Earth's orbital velocity) as compared to velocity of neutrino in the Galaxy. The role of slow component in effects of accumulation and annihilation of 4th generation neutrino in the Earth will be considered in separate work. Distribution of Sun's matter density with the solar radius x is supposed in accordance with standard solar model [12] and it can be approximated by function p(x) = 148(1 - x2) exp ( - ^ ) g/cmJ. (5) s 0 1 1 5 2 The x dependence for square velocity v^ (x) is well approximated by function scS 1 - z1-868 + 1 ] , (6) + • where v s = v s(l) = 618 km/s. The values of parabolic and maximal total velocities esc osc for neutrinos in Sun's centre and of kinetic energies corresponding to them are respectively equal to v (0) = 1383 km/s, v{0) = ^/v^+ v (0) = 1416 km/s, T (0) = 513 keV, escS s cscS oscS Ts(0) =T + T (0) = 556 keV. OO escS Now let us consider cross sections of interaction of massive neutrino with nuclei. The wavelength of neutrino with the mass 50 GeV and the velocities 300 4-1416 km/s are correspondingly A = (3.9 -4- 0.84) • 10~13 cm. That is comparable with nucleus sizes. Therefore it is necessary to take into account the effects of the finite size of nuclei, i.e., that nuclei are not point-like. For all nuclei presented in the considerable amount inside the Sun and Earth the lowest excited energies (> 0.8 MeV) exceed the kinetic neutrino energy (< 0.56 MeV). Therefore the scattering of a neutrino on a nucleus we will consider as the elastic one on the whole nucleus. The amplitude of scattering of a neutrino (antineutrino) on a nucleus in this case is determined by coherent isoscalar vector neutral weak coupling and in the nonrelativistic limit is given by expression [13] where G is Fermi's constant; F = '1Z - A - AZ sin2 0 = -(A - 1.074Z); A, Z are the F v W atomic number and the charge of the nucleus; #w is the Weinberg angle; F{q2) is the form factor, taking into account nucleus non-point-likeness. The cross section will be where fj, = is the reduced mass of neutrino m and nucleus m^; (T) is a factor, u m,, + m.A taking into account nucleus non-point-likeness and dependent on kinetic energy (velocity) of Monochromatic Neutrinos from Massive Fourth 9 a neutrino. For proton F + V « 0, and the axial coupling plays the main role in interaction with a neutrino. The cross section of the proton-neutrino interaction in nonrelativistic limit takes the form ^ + FlUT) = aoniT). (7) For a proton (as well as for a neutron) FA « 1-25. The dependence of form factor on q2 is assumed in accordance with the model of nucleon Fermi gas for all nuclei. That, generally speaking, is not quite correct for light nuclei. Since the neutrino energy is not enough for the excitation of nuclei, the nucleon Fermi gas is considered as strongly degenerated. In this case ^ ^ y where q is the modulus of neutrino 3-momentum transferred to nucleus. The factor rj(T) will have the form _ 9 -1 + cos 2y - 2y? + 2y sin 2y + 2j/,4 max riax max m&x nax ^ i/max (g) 2/max — Qma.x^' — with limits 1 for j/2 < 1 (or v2 «; vl ), iax ohn —- for yl, » 1 (or v2 » v2 r ax VQ AHmax where (3 = i;/c is the neutrino velocity (nucleus velocity is neglected everywhere); c is the velocity of light; v mt = 7.— is the characteristic minimal velocity at which the nucleus pO JjJLCl cannot be considered point-like. The nucleus size is a = 1.25 fm^41//3 158 MeV It is worth to note the important role of heavy nuclei in massive neutrino capture. Firstly, the cross section rises strongly with increasing atomic number, for TUA <C rn, v2 <g; Wp ; it v Ont behaves as A4. Secondly, for nuclei with the mass close to 50 GeV for kinematics reasons the probability of greater energy loss by neutrino in one collision increases. Correspondingly for them the probability of neutrino capture in one collision wi for that the neutrino has to lose the energy Too = 25 keV will be greater. The value wi is defined by probability distribution in neutrino energy transferred to the nucleus. The form of such distribution depends on nucleus non-point-likeness: it has «table-like» form for point-like nucleus and in the opposite case, when the non-point-likeness is essential, falls down with the energy transfer. In the general case w\ can be written as /(j/max) 10 Belotsky KM., Khlopov M.Yu., Shibaev K.I. where y^ =. /m mA^»fl, A» = Voa/c Function f(y) is contained in r] (see (8)) and has A o ( the form _ -1 + cos 2y - 2z/2 + 2y sin 1y + 2t/4 2y with the limits f{y) « -J- for 2/ « 1 and y ^ /(y) ~ 1 r- for y <C 1 (the small term 1/y2 is important in the case y2 2/m<ax » 1 and t/oo > 1); w\ is meaningful for yoo > y or max t> < -yv-infty, 7 = (11) (t> = v^oo+^ScO^))- ^n me case °f ^e Earth the neutrino can be captured in collision with nuclei with the mass close to 50 GeV (46 -=- 54 GeV) only. In this mass interval the nucleus of iron, which is abundant in the Earth, hits successfully. The probability W\ reaches in this case ~ 10~3. In the case of the Sun the inequality (11) holds for all nuclei except hydrogen. The probability w\ for hydrogen (proton) is positive just in the region inside the Sun with xe [0;0.29]. Using Eqs. (8), (9) for 77 and wi, expression (3) can be transformed to Ncapt = > O-oiV^tot . (12) ~T Voo J» A tot Here NAtot = -—4—NA is the total number of nuclei A in gravitating body; (a^) is the mass fraction of element A in the whole gravitating body with mass M; N is Avogadro's A number; nA = —J-NA is the number density of nuclei A. In the limiting cases formula (12) can be simplified. 4 2l for v2 « ^ oillt> In Eq. (13) (v2 ) means the square escape velocity averaged over the (number) density of sc nuclei A in the Earth or the Sun. In the case of homogenous relative abundance of element A over the volume of the gravitating body, i.e., when aA — const = {aA), (V2SC) is reduced to square escape velocity averaged over total density f v2 (r\nHV WJSC/ M In this case for chosen density distribution of the Earth we have (V^SCE) = "^cscE- F°r ^e 5 Sun one obtains {^ } ~ 3.3^ . scS scS Monochromatic Neutrinos from Massive Fourth 11 In another limiting case Eq. (12) has the form iV = J2 ^^N tot|^oint(l " /(Woo)) for v2 » t£ . (14) capt A oint A Vo° Z All notions being used in Eqs.(12)-(14) are introduced above. The data on chemical composition of Earth's interiors, in which we are first of all interested in the iron abundance, are virtually absent. It is just known that the iron is likely to predominate in Earth's core and that it is contained in considerable amount in the mantle. We put iron mass fraction in the Earth equal to (ap ) = 20 % and we will consider its abundance e homogeneous inside Earth's volume. The account of the increase of iron number density towards Earth's centre would lead to small increase of the result. For simplicity we will assume the solar chemical composition homogeneous too and take it from the observation of solar atmosphere. The calculation based on the standard solar model shows, generally speaking, the increase of the helium fraction, decrease of the hydrogen fraction and radial variability of abundance of other elements. The account of such inhomogeneity would lead to small increase of the result mainly due to the increase of helium fraction. 2. ACCUMULATION AND ANNIHILATION OF MASSIVE NEUTRINOS IN THE EARTH AND SUN Let us consider now the processes of accumulation and annihilation of neutrinos. To calculate the rate of neutrino capture by the Earth we take into account n « n, •oo> v « VOQ. It is convenient to use the expression (3) and to factor out the integral sign everywhere except W\. Then this formula will be analogous to Eq. (13) with the additional factor r) denned by (8) and we obtain 77^0.86, _ ^ - > x 17 5 Ncapt = nooUooCTo^Fetot^l) ~ 1-0 • 1014 S"1. The estimations show that the contribution of multiple collisions to neutrino capture does not exceed 10%. Also the account of neutrino velocity distribution is not significant. Indeed, if the neutrinos have Maxwellian distribution with the mean velocity 300 km/s, then the result rises in 1.3 times. If we take into account in this distribution the motion of solar system (relative to galactic centre) with the velocity 200 km/s, then the result rises in less than 10 %. We do not expect a large difference in correction, connected with velocity distribution of neutrinos, for the case of the Sun. The neutrinos, captured by Earth's field, during the time (of the order of a year) much less than Earth's age lose in collisions with the matter their energy down to thermal energy of nuclei and concentrate in Earth's core. We will further refer to this process as to the thermalization and to its characteristic time as to the time of thermalization. In order to estimate the amount of neutrino accumulated in the Earth, for simplicity we will assume that 12 Belotsky KM., Khlopov M.Yu., Shibaev K.I. the concentration of thermalized neutrinos is distributed homogeneously inside the sphere of radius Xtiierm- This radius is determined as the maximal distance at which a particle can be with total energy equal to Therm + U(0), where T], rm and f/(0) are thermal energy of t t C particles (molecules) and potential energy of neutrino in Earth's centre. The temperature in Earth's centre is not known exactly and it is supposed in the range 3000 -f- 10000 K. We will put Tthcrm = 1 eV- The value of x\ is obtained with the account of Eq. (4) from the tKlin equation „-, _ 3i E _ _rp (3 — X ) csc therm J- therm „ ~ -^escE „ ' 2 ^cscE = —" cscE eV, and is equal to £hc.m ~ 0.24. The accumulation of neutrinos inside o t the Earth (growth of thermalized neutrino concentration) is stopped when the rate of their annihilation becomes equal to capture rate 7V . = iV t (the energy realization from anneq cap annihilation in this case will be 0.84 • 1013 erg/s that corresponds to 0.5 • 1012 of the total energy flux incident on the Earth from the Sun). Note that the account of real thermodynamic distribution of thermalized neutrinos must lead to additional process of evaporation (escape out the Earth). But as it was shown in [9] for WIMP mass greater than about 10 GeV the annihilation rate significantly exceeds the evaporation one. So our approximation is in principle valid. The annihilation cross section of neutrinos is determined by weak neutrino interaction near Z-boson resonance. In nonrelativistic approximation the cross section of such annihilation into pair of neutrino and antineutrino of certain kind is t ml 1 1.28 • 10~34 cm2 O-ann.into^ = ^ ( 2 _ | )2 ^ ~ J* " 4m m And the total cross-section of annihilation is equal to _ ffaiin.into^g. ^ 1.93 • 10~33 Cm2 <7a"" ~ Br(Z -» vv) ~ 13* c c where gJA\f2G-pm2 is the dimensionless constant of the weak interaction; Gp is the z Fermi constant; mz is the iv-boson mass; /?* = v*/c; v* is the velocity of neutrino V4 in the centre of mass system; Br(Z —> vv) « 6.67% is the branching ratio of the e e Z-boson decay into neutrino and antineutrino pair of certain kind. The equilibrium con- centration n and the corresponding total amount JV of thermalized neutrinos will be oq cq (•Afann.eq = («e /4)cr wciVthcnn, where v \ = \/2v* is the mean relative velocity of q aim r vc neutrinos, Vti, — -7r(i?Ea;tiion!i)3 « 1.5 • 102° cm3) eim n w 0.58 • 106 cm"3 cq N q = ncqVthorm ~ 0.88 • 1031 (corresponds to mass of 780 tons). C The time necessary to establish such equilibrium is equal to N Monochromatic Neutrinos from Massive Fourth 13 The obtained time is comparable with Earth's age (it is only 2 times smaller), therefore the current (accumulated for Earth's lifetime) amount of neutrinos N can differ from JV . cq Extracting N from equation N = iV — N we obtain capt ann N = N tanh - 0.964N , eq eq 'eq where t& is modern Earth's age. Correspondingly iV will be ann N = 0.9642iV = 0.93iV . mn capt capt In our approximation we can assume TV ~ N , iV sa iV . eq ann capt 20 % of neutrinos i> annihilated via intermediate Z boson produce monochromatic neu- 4 trinos of known types (vv, v v v v ) with the energy of 50 GeV. Their flux on Earth' e c M fl T T surface will be j — The partial flux for each neutrino type («neutrino + antineutrino») is 3 times smaller. For the Sun the neutrino capture rates on concrete nuclei iv~ p. n A are given in Table 1 ca t O (the elements are arranged in the order of their concentration decrease in the Sun, the data about chemical composition were taken from [14]). Table 1. Element A^capt.on^, 1020 S-1 Element Nce.pt. on A, 102° S""1 Element Ncs.pt. on A, 1020 S"1 jH 0.94 26 Fe 3.43 uSi 1.27 2He 4.60 IGS 0.75 24 Cr 0.05 o 5.89 isAr 0.29 17CI 0.02 8 6c 1.62 13AI 0.11 15P 0.01 ioNe 1.94 2oCa 0.15 25M11* 0.03 7N 0.53 uNa 0.05 19K 0.01 12Mg 0.97 28Ni 0.18 22Ti 0.01 14S1 1.27 24Cr 0.05 27CO 0.01 "The lowest excited level of nucleus 25M1155 is 126 keV. By interacting with neutrino with the energy (inside the Sun) of 131 -=- 556 keV the inelastic scattering is possible. Here we did not taken it into account. While calculating interaction cross-section the formula (7) was used. The total rate of neutrino V4 capture by the Sun is N, 2.2-1021 s"1. capt The total neutrino v4 flux through the full solar surface is nasv = n, v TTRS sa 4.8-10•^ 23 sc-l Here n and v are number density and velocity of neutrinos on the (level of) solar surface (v = 637 km/s), as = 7ri?| is Sun's cross section. One can see that the incident neutrino flux 14 Belotsky KM., Khlopov M.Yu., Shibaev K.I. is much greater than iVc . Therefore the approach, in which the change of incident neutrino apt concentration induced by interaction with a matter is neglected, is valid. Also it should be noted that the relation between 7v* and incident flux together with the applicability of our capt approach does not depend on rioo- The thermalization time of neutrinos captured by the Sun is much smaller than Sun's age. It is of the order of a year as well as for the Earth. Neutrino kinetic energy in solar centre will be Ttii = 1.9 keV, potential energy is obtained from expression (6) by multiplying it Onn by the factor - ^ / 2. Then x\ « 0.017, Vtherm « 0.69 • 1028 cm3, n « 1.3 • 108 cm"3, tKrm eq N w 0.87 • 103G (that corresponds to mass 0-78 • 1014 g), t « 1.2 • 107 y, that means cq eq N = N , Nann = N . The energy realization from -y annihilation is 1.8 • 1020 erg/s = cq Ciipt 4 4.7 • 1014Ls, where L = 3.8 • 1033 erg/s is the luminosity of the Sun. s The 50 GeV annihilational neutrinos of known types will be partially captured inside the Sun. The cross section of neutrino and antineutrino (with the energy of 50 GeV) interaction with nucleon differ from each other and are [15] 5.1 • 1037 cm2 and 2.6 • 103' cm2, respec- tively. Note the difference of cross sections for different kinds of annihilational neutrinos is negligible. The fractions of neutrinos and antineutrinos escaping from the Sun will be respec- tively 0.63 and 0.79. The fluxes on the Earth of monochromatic neutrinos, antineutrinos and summary («neutrinos + antineutrinos*) from ^annihilation in the Sun will be I = 0.50-107 cm^-s"1, v h =0.63-107 cm"2-iT1, = l.l-107cm -2-s-1 u5) For each neutrino kind these fluxes are correspondingly 3 times smaller. The most uncertain parameter in our model is the neutrino number density n^, which is proportional to ^Q.. The values N , I and equilibrium time t depend on the number cilpt cq density n<x>- The results are generalized for the case of arbitrary £s^ by the following way j « 1.1. lO"7 (^pl tanh2 (J gs" ) cm"2 • s"1 - for the Sun. For the Earth, generalizing also the results for the case of arbitrary mass fraction of iron, we have Here £s^ = 2 corresponds to the magnitude used in this work (£fi = 2 • 104 • 10~4 = 2, s rice = 2 • 10~7 cm"3), tanhx « 1 for x > 1. - The solid angle of the annihilational neutrino flux from Earth's core is about lsr, the one for the Sun will be determined by angle resolution of detector. The fluxes of atmospheric neutrinos depend on direction especially for high energy electronic neutrinos [16]: for hor- izontal direction the fluxes are greater than for vertical one. The comparison of fluxes of monochromatic annihilational neutrinos with the atmospheric ones in vertical direction for different energies is presented in Table 2. In the first column the ratios of considered fluxes